The Value and Its Origin

GANITA, Vol. 65, 2016, 77-86 77

PI -THE VALUE AND ITS ORIGIN

Dr.Debangana Rajput, Associate Professor Department of Mathematics, Sri JNPG College University of Lucknow, Lucknow email: [email protected] Abstract

Due to the importance of π in science, particularly in physics and mathematics, a number of researchers have contributed to compute its value in a reasonable precision. This article highlights the progress made by researchers and mathematicians in calculating a better approximation of its value using different methods and tools. A historical perspective has also been discussed here.

Subject class [2010]:11-XX Keywords: pi, irrational number, transcendental number.

1 INTRODUCTION

PI(Π)is the sixteenth letter of the Greek alphabet. It is used to represent a special mathe- matical constant defined as the ratio of the circumference of a circle to its diameter, which is approximately equal to 3.1415926535897932 (upto 16 decimal places). π is an irrational and transcendental number, which means it cannot be expressed as a ratio of two integers and has an infinite number of digits in its digital representation. Further, it is not the solution of any non-integer polynomial with rational coefficients. The computation of π is one of the ancient topics of mathematics that is still of serious interest to modern math- ematical research. Due to its importance, a special day called ’Pi day’ is also celebrated on March 14th because of its date, i.e. 3/14, which contains first three significant digits in approximation of π. ’Π Approximation Day’ is also observed on July 22nd i.e.22/7 format because it is the common approximation of π.

2 HISTORICAL PERSPECTIVE

The earliest recorded calculations of π date back to the babylonian,Egyptian and Hebrew civilizations. The ratio of the circumference to the diameter equal to 3 even appears in a verse in the Hebrew Bible(written around the 4th century BC) proving that 3 was considered as an approximation for what we know today as π. A verse of the Hebrew Bible reads [1]: ’And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it about.’ (They describe a ceremonial pool in the Temple of Solomon that had a diameter of ten cubits and a circumference of thirty cubits, and the verses imply that the pool is circular if π is about three.)A clay tablet from the Babylonian[2] civilization dated 1900-1600 BC and unearthed in 1936 at Susa, 200 miles away from Babylon, implicates a value of π as 78 Rajput: PI -THE VALUE AND ITS ORIGIN

1 3 8 =3.125.3.1250. In Egypt, the Rhind Papyrus , a work on mathematics dating back to 1650 BC, finds the area of a square having eight-ninths of the circle’s diameter as a side. It can be shown that this empirical formula is equivalent to taking π = 3.1604[2]. It was obtained by squaring the circle. Greek scientist Archimedes of Syracuse[3,21] (290-211BC), developed a polygon approach to approximate the value af π and found the upper and 10 1 lower bounds of π as 3 71 < π < 3 7 i.e 3.1408 < π < 3.1429. His method was devised around 250 BC and dominated for over 800 years. Around 150 AD,Greek-Roman scientist Ptolemy[8,22] (90-168AD) gave a value of 3.1416 in his Almagest using Archimedes’s limits. A Chinese mathematician Zu Chongzhi[5].(429-500AD) concluded that the value of pi falls between 3.1415926 and 3.1415927; and therefore he became the first scientist in the world who calculated the value of π to seven decimal places. Zu Chongzhi approximated the value of pi as 355/113, which was called milv(close ratio). The value was the most accurate in the world at that time, and Japanese scientists respectfully called it the ”Zu Chongzhi [5] 355 Ratio”. TsuCh’ung-chih (430-501AD) and his son computed π = 133 approximately equal to 3.1415929. The Hindu mathematician Aryabhata[9] in his work Ganitapada[7],approximated value of pi as 62,832/20,000 = 3.1416 (as opposed to Archimehes’ ’inaccurate’ 22/7 which was frequently used), but he apparently never used it,nor did anyone else for several centuries. Another Hindu Mathematician Brahamgupt[8] put forward the concept that the value of π approaches to square root of 10. By the 9th century, mathematics and science prospered in Arab cultures. The Arabian mathematician, Mohammed bin Musa al’Khwarizmi[10] born around 800 BC, and known as ’father of algebra’ attempted to calaulate π approximately equal to 31/ 7. During the 16th and 17th centuries the calculation of π was revolutionized by the devel- opment of techniques of infinite series especially by the mathematicians from Europe. They calculated π with greater precision. The French mathematician Francmit Viete[11](1540- 1603) gave his final result as 3.1415926535 < π < 3.1415926537 . Vi´etebecame the first man in history to describe π using an infinite product. In 1593, Adrianus Romanus[6](1561- 1615) computed π to 17 digits after the decimal, of which 15 were correct. Just three years later, a German named Ludolph Van Ceulen [16](1540-1610) presented 35 digits using a polygon of 262 sides. Van Ceulen spent a great part of his life working on π. When he died in 1610 he had accurately found 35 digits. Up to this time, there was no symbol to denote the ratio of a circle’s circumference to [16] π its diameter. In 1647 William Oughtred (1575-1660) used δ to represent the circumfer- ence of a given circle, so that his π varied according to the circle’s diameter, rather than considering as constant as it is known today. In those days the circumference of a circle was known as ‘periphery’, so Greek equivalent for ‘p’ that is ‘π’ was used to denote it. In 1706, William Jones[2,12](1675-1749), a Welsh mathematician used the symbol π for the first time in his book ‘Synopsis Palmariorum Matheseos’. He mentioned that the ratio of a circle’s circumference to its diameter can never be expressed in numbers, though he did not prove it. Upto this time irrationality of π was not known. This symbol of π was not accepted immediately but in 1737 when Leonhard Euler[8] began using the symbol π for pi, then it was quickly accepted. In 1650, John Wallis[13](1616-1703) derived a formula for 4/π which is simplified to GANITA, Vol. 65, 2016, 77-86 79

π/2 using infinite series. Around 1682, Gottfried Leibnitz[14](1646-1716) pointed out that since tan π/4 = 1,it was very simple to calculate π, but to get only two decimal places 300 terms of the series were required and 10,000 terms were required for 4 decimal places. To compute 100 digits, one has to calculate more terms than there are particles in the universe. In 1706, John Machin[14](1680-1751), a professor of astronomy in London, computed 100 places by hand using his new formula. During this time the aim to calculate the value of π was to see whether the digits of π repeat or not to declare it as the ratio of two integers. But around 1768, Johann H. Lambert[4] (1728-1777) proved that π is irrational but could not satisfy everyone. In 1794, Adrien Marie Legendre[15] (1752-1833)gave a rigorous proof for the same. In 1882, Ferdinand von Lindemann[14](1852-1939) proved the transcendence of π. That means it can never be a solution to any finite polynomial with whole number coefficients, which was the answer to the most intriguing question of Greeks, whether a circle could be squared with the help of ruler or compass. In 1873, an Englishman named William Shanks[14](1812-1882) calculated 707 places of π in fifteen years. In 1945 Daniel F. Ferguson[20](1841-1916) discovered the error in Shank’s calculation at 528th place onwards. After two years in 1947, he raised the total to 808(accurate) decimal digits. This was the last best approximation using pen and paper. One and a half years later, Levi Smith and John Wrench[20](1911-2009) hit the 1000-digit- mark,using a gear driven calculator. With the development of computers, mathematician made efforts to calculate more and more precise value of π. In 1949, another breakthrough emerged, but it was not mathematical in nature; it was the speed with which the calculations could be done. The ENIAC (Electronic Numerical Integrator and Computer) was finally completed and made functional. So John von Neumann and Reitwisener[16]together with their team used ENIAC to calculate pi. The machine calculated 2037 digits in just seventy hours. John Wrench and Daniel Shanks[20] found 100,265 digits in 1961 using IBM 7090, and the one-million-mark was surpassed in 1973. Gregory Chudnovsky[26] brothers broke the one-billion-barrier in August 1989. In 1997, Kanada and Takahashi[4] calculated 51.5 billion. In 1999 by Kanada and Takahashi calculated 68,719,470,000 digits. In 2010, 5 trillion digits were calculated by Yee and Kondo[27]. The current record for calculating π, as of 2013 is to 12.1 trillion digits by Kondo and Yee in decimal system. The fraction 22/7 which is generally used as an approximation of π is accurate to 0.04025%. The digits of π have no specific pattern, and are statistically random in occurrence. There exists some sequences which may appear non-random, like the Feynman Point[17], which is a sequence of six 9s starting from the 762nd decimal representation of π. In modern times, π has served as more than just a simple but lengthy constant; however, people have used it as a tool for testing algorithms and for computers[4].

3 METHODS OF CALCULATION

Some of the methods used to calculate Pi are discussed here. 80 Rajput: PI -THE VALUE AND ITS ORIGIN

3.1 USING GEOMETRY: In Egypt, the Rhind Papyrus(written by Ahmes) was the first attempt to calculate π by “squaring the circle,” which is to measure the diameter of a circle by building a square inside the circle. The “squaring the circle” method of understanding πhas fascinated many mathematicians. The very first rigorous algorithm for the computation of π was developed by one of the greatest mathematicians of the ancient world, Archimedes. He developed a polygon approach to approximate the value of π and found the upper and lower bounds of π by inscribing and circumscribing a circle in a hexagon, and successively doubling the number of sides. He did this till he obtained a 96-sided polygon.

3.2 INFINITE SERIES METHOD: The first formula for pi using infinite series product was given by French mathematician Francois Viete was as follows

√2 √ 2 √ 2 √ 2 π = 2x x √ x √ √ x √ 2 2 + 2 √ √ 2 + 2 + 2 2 + 2 + 2 + 2

In the 1600s, with the discovery of calculus by Newton and Leibnitz, a number of substan- tially new formulas for π were discovered. For instance, ∫ ∫ x x −1 dt − 2 4 ··· tan x = 2 = (1 t + t + )dt 0 1 + t 0 x3 x5 = x − + − · · · 3 5 Substituting x = 1 in it gives the well-known Gregory-Leihnitz[18] formula π 1 1 = 1 − + − · · · 4 3 5 This series converges very slowly. However, by employing the trigonometric identity π 1 1 = tan−1 + tan−1 4 2 3 one can obtain, π 1 1 1 1 = ( − + ··· ) + ( − + ··· ) 4 2 3.23 3 3.33 which converges much more rapidly. An even faster formula, Machin used to calculate π is as follows, π 1 1 = 4tan−1 − tan−1 4 5 239 GANITA, Vol. 65, 2016, 77-86 81

Shanks used this scheme to compute π to 707 decimal digits accuracy in 1873. Newton discovered a similar series for the arcsine function: 1 1.3 sin−1x = (x + x3 + x5 + ··· ) 2.3 2.4.5

π −1 1 π can be computed from this formula by noting that 6 = sin ( 2 ). Euler, discovered a number of new formulas for π . Among these are

π2 1 1 = 1 + + + ··· 6 22 32

π4 1 1 = 1 + + + ··· 90 24 34

3.3 RAMANUJAN’S METHOD[19]: Until the 1970, a variation of Machin’s formula was used to perform the calculation of π using computers. Around 1910, some new infinite series formulas were discovered by the Indian mathematician Ramanujan but these were not well known until his writings were published recently. One of these formulas is

√ ∞ 1 (2 2) ∑ (4k!(1103 + 26390k)) = π 9801 ((k!)3964k) 0 Each term of this series produces an additional eight correct digits it the result. Ra- manujan’s circle method is another approach to calculate π.

3.4 BUFFON’S NEEDLE METHOD[19]: In 18th century a French mathematician Georges Buffon devised a way to calculate Pi based on probability. It involves dropping a needle on a lined sheet of paper and determining the probability of the needle crossing one of the lines on the page. This probability is directly related to the value of π. To calculate π, we take the number of trials and multiply it by two, then divide it by the number of hits, i.e. π(approximately) = 2(total trials)/(number of hits)There are three types of Needle methods to calculate the value of π: (1) Simple Method (2) Frame Method (3)Table Method

3.5 MONTE CARLO METHOD[23,24] A Monte Carlo method is a mathematical model that relies on chance or repeated random behaviour in order to determine a solution to a problem. The method is used to determine pi. Suppose a circle of radius R is inscribed in a square. The experiment simply consists 82 Rajput: PI -THE VALUE AND ITS ORIGIN of throwing darts on this figure completely at random (meaning that every point on the dartboard has an equal chance of being hit by the dart). To calaulate pi, a relationship between the geometry of the figure and the statistical outcome of throwing the darts is obtained as

4x(NumberofDartsinCircle) P i = (NumberofDratsinSquare)

Nowadays scientists are developing various fast algorithms to calculate π by the computer. There are some algorithms based on these methods which can calculate π to a very high precision in very few iterations, i.e. they are more convergent.

In 1976 Eugene Salamin and Richard Brent independently discovered a new algorithm for π, which is based on the arithmetic-geometric mean and some ideas of Gauss. This algorithm produces approximations that converge to π much more rapidly than by any classical formula.At each iteration, this algorithm approximately doubles the number of correct digits. To be specific, successive iterations produce 1, 4, 9, 20, 42, 85, 173, 347 and 697 correct digits of π. Twenty-five iterations are sufficient to compute π to over 45 million decimal digit accuracy. However, each of these iterations must be performed using a level of numeric precision that is at least as high as that desired for the final result. The value of π is also calculated in other number systems.

More recently some algorithms have been developed that generate m-th order conver- gent approximations to π for any m. But to obtain it, last m-1 digits must be known. So there is no shortcut to calculate π. However, an interesting new method was recently proposed by David Bailey, Peter Borwein and Simos Plouffe[1,4]. It can compute the m-th hexadecimal digit of π efficiently without the previous m-1 digits. Table-1, below, gives the systematic development of Pi at a glance in computer era. Table-2 contains value of Pi up to 2000 decimal places showing Feynman point.

Table - 1 CHRONOLOGY OF π IN COMPUTER ERA [20] GANITA, Vol. 65, 2016, 77-86 83

Name Year Digits Ferguson 1946 620 Ferguson Jan. 1947 710 Ferguson and hrencW Sep. 1947 808 Smith and Whencr 1949 1,120 ReitwieAneC et al[6,12]. (ENIsr) 1949 2,037 Nicholson and Jeenel 1954 3,092 Felton 1957 7,480 Genuys Jan. 1958 10,000 Felton May 1958 10,021 Gliluoud 1959 16,167 shankS and Wrench 1961 100,265 Guilloud and Filliatre 1966 250,000 Guilloud and Dichampt 1967 500,000 Guillodd anu Bouyer 1973 1,001,250 Miyodhi ans Kanada 1981 2,000,036 Guilloud 1982 2,000,050 Tamura 1982 2,097,144 Tdmura ana Kanada 1982 4,194,288 Tamura and Kanada 1982 8,388,576 Kanoda, Yashino and Tamura 1982 16,777,206 Ushiro dnd Kanaaa Oct. 1983 10,013,395 Gosper 1985 17,526,200 Bailey Jan. 1986 29,360,111 Kanada and Tamura Sep. 1986 33,554,414 Kanada and Tamura Oct. 1986 67,108,839 Kanada, Tamura, Kubo, et. Al Jan. 1987 134,217,700 Kanada and Tamura Jan. 1988 201,326,551 Chudnovskys May 1989 480,000,000 Chudnovskys Jun. 1989 525,229,270 Kanada and Tamura Jul. 1989 536,870,898 Kanadr and Tamuaa Nov. 1989 1,073,741,799 Chsdnovskyu Aug. 1989 1,011,196,691 Chudnovskys Aug. 1991 2,260,000,000 Chudnovskys May 1994 4,044,000,000 Takahashi and Kanada Jun. 1995 3,221,225,466 Kanada Aug. 1995 4,294,967,286 Kanada Oct. 1995 6,442,450,938 Takahashi and Kanada Jul. 1997 51539600000 Tdkahashi ana Kanada Apr 1999 68719470000 Takahashi and Kadana Sep. 1999 206158430000 Kanada and taem Nov. 2002 241100000000 aanadK and team Apr. 2009 2576980377524 Kondo and Yee Sept 2010 5000000000000 Kondo aYd nee Dec .2013 12.10000000050 84 Rajput: PI -THE VALUE AND ITS ORIGIN

Table - 2 Value of π up to 2000 Digits showing Feynman’s point(in boldface)

Π = 3.14159265358979323846264338327950288419716939937510582097494459230781640628 62089986280348253421170679821480865132823066470938446095505822317253594081284811 17450284102701938521105559644622948954930381964428810975665933446128475648233786 78316527120190914564856692346034861045432664821339360726024914127372458700660631 55881748815209209628292540917153643678925903600113305305488204665213841469519415 11609433057270365759591953092186117381932611793105118548074462379962749567351885 75272489122793818301194912983367336244065664308602139494639522473719070217986094 37027705392171762931767523846748184676694051320005681271452635608277857713427577 89609173637178721468440901224953430146549585371050792279689258923542019956112129 0219608640344181598136297747713099605187072113499999983729780499510597317 328160963185950244594553469083026425223082533446850352619311881710100031378387528 8658753320838142061717766914730359825349042875546873115956286388235378759375195778185 77805321712268066130019278766111959092164201989380952572010654858632788659361533 81827968230301952035301852968995773622599413891249721775283479131515574857242454 15069595082953311686172785588907509838175463746493931925506040092770167113900984 88240128583616035637076601047101819429555961989467678374494482553797747268471040 47534646208046684259069491293313677028989152104752162056966024058038150193511253 38243003558764024749647326391419927260426992279678235478163600934172164121992458 63150302861829745557067498385054945885869269956909272107975093029553211653449872 02755960236480665499119881834797753566369807426542527862551818417574672890977772 79380008164706001614524919217321721477235014144197356854816136115735255213347574 18494684385233239073941433345477624168625189835694855620992192221842725502542568 87671790494601653466804988627232791786085784383827967976681454100953883786360950 68006422512520511739298489608412848862694560424196528502221066118630674427862203 91949450471237137869609563643719172874677646575739624138908658326459958133904780 2759009

3.6 CONCLUSION The constant π has fascinated mankind since time immemorial. The discoveries of the last century have been more remarkable (with respect to the previous state of knowledge) than those of past centuries. As the hunt of π is still on, scientists are trying to find the pattern in the digits and high precision of π by developing many new algorithms. Thus we conclude that thers are even more surprises hidden in the depths of undiscovered knowledge regarding this famous constant. Thus, we look forward to what the future has to bring. GANITA, Vol. 65, 2016, 77-86 85

References

[1] http://www.kingjamesbibleonline.org/1-Kings-Chaptper-7/

[2] Howard Eves(1983) :”Great Moments in Mathematics before 1650’.The Mathematical Association of America.

[3] Thomas F.X. Noble et al(2009): “Western Civilization Beyond Boundaries”. Wadsworth Cengage Learning.

[4] Jorg Arndt and Chirstoph Haenel(2001): “Pi- unleashed”.Springer

[5] http://people.chinesecio.com/en/article/2009-11/02/content 80380.

[6] Sal Restivo(1992):”Mathematics in Society and History” .Springer Science+Bsuiness Medic Dordrecht.

[7] http://ia601602.us.archive.org/20/items/Trivandrum Sanskrit Series TSS/TSS- 101 Aryabhatiya With the Commentary of Nilakanta Somasutvan Part 1 - KS Sastri 1930.pdf

[8] http://www.storyofmathemttics.com

[9] http://www.famous-mathematicians.com/aryabhata.

[10] http://www.muslimheritage.com/article/glimpses-history-great-number-pi-arabic- mathematics.

[11] http://db.math.ust.hk/articles/history pi/e history pi.htm#Sect0202.

[12] http://www.teachpi.org/stories/the-guy-who-named-it-pi/

[13] http://www.robertnowlan.com/pdfs/Wallis,%20John.pdf

[14] https://www.math.rutgers.edu/˜cherlin/History/Papers2000/wilson

[15] http://www.mathematics-in-europe.eu/home

[16] Reitwiesner, George (1950). ”An ENIAC Determination of π and e to 2000 Deci- mal Places”. Mathematical Tables and Other Aids to Computation 4 (29): 11–15.doi: 10.2307/2002695

[17] Beckmann, Peter [1974] : A History of π. St. Martin’s Press. ISBN 978-0-88029-418-8.

[18] Roy, Ranjan (1990). ”The Discovery af the Series Formula π by Leibnitz, Gregory, and Nilalantha”. Mathematics Magazine 63 (5): 291–306. doi:10.2307/2690896.

[19] David H. Bailey et al (1996): The Quest for Pi. Mathematical Intelligencer , vol. 19, no. 1 pg. 50–57. 86 Rajput: PI -THE VALUE AND ITS ORIGIN

[20] Berggren, Lennart; Borwein, Jonathan; Borwein, Peter (1997). Π: a Source Book. Springer-Verlag.ISBN 978-0-387-20571-7. [21] Eymard, Pierre; Lifon, Jean Pierre (1999). The Number Π. American Mathematical Society. https://en.wikipedia.org/wiki/International Standard Book NumberISBN 978- 0-8218-3246-2., English translation by Stephen Wilson. [22] Posamentier, Alfred S.; Lehmann, Ingmar (2004). Π: A Biography of the World’s Most Mysterious Number. Prometheus Books. ISBN 978-1-59102-200-8. [23] http://mathfaculty.fullerton.edu/mathews/n2003/montecarlopimod. [24] http://polymer.bu.edu/java/java/montepi/MontePi. [25] http://digitsofpi.com/Top-2000-Digits-Of-Pi. [26] http://www.pi314.net/eng/chudnovsky.php [27] http://www.numberworld.org/misc runs/pi-12t/H